Results 1 to 6 of 6
Like Tree4Thanks
  • 1 Post By MarkFL
  • 2 Post By Prove It
  • 1 Post By MarkFL

Math Help - Evaluate the derivative using properties of logarithms where needed.

  1. #1
    JDS
    JDS is offline
    Member
    Joined
    Oct 2012
    From
    South Carolina
    Posts
    83

    Evaluate the derivative using properties of logarithms where needed.

    As some added context, I am taking Calculus: Early Transcendental Functions and we are studying "The Natural Logarithm as an Integral"

    The problem presented is as follows:

    d/dx [ln (x5 sin x cos x)]

    I've read through the material and even looked at some of the odd numbered problems that have answers but I just can't seem to get started here. Any advice on how to approach this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Evaluate the derivative using properties of logarithms where needed.

    I think I would simplify the application of the chain rule a bit by first writing the expression as:

    \frac{d}{dx}\left(\ln(x^5\sin(2x))-\ln(2) \right)

    Now, the derivative of the constant \ln(2) is zero, so we are left with:

    \frac{d}{dx}\left(\ln(x^5\sin(2x)) \right)

    See if you can now apply the rule:

    \frac{d}{dx}\left(\ln(u(x)) \right)=\frac{1}{u(x)}\cdot\frac{du}{dx}
    Thanks from JDS
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,409
    Thanks
    1294

    Re: Evaluate the derivative using properties of logarithms where needed.

    Quote Originally Posted by JDS View Post
    As some added context, I am taking Calculus: Early Transcendental Functions and we are studying "The Natural Logarithm as an Integral"

    The problem presented is as follows:

    d/dx [ln (x5 sin x cos x)]

    I've read through the material and even looked at some of the odd numbered problems that have answers but I just can't seem to get started here. Any advice on how to approach this?
    Following Mark's example of simplifying the logarithm, you should simplify even further before trying to take the derivative. Write it as \displaystyle \begin{align*} y = 5\ln{(x)} + \ln{\left[ \sin{(x)} \right]} + \ln{\left[ \cos{(x)} \right]}  \end{align*} and then apply the much simpler chain rules to each term.
    Thanks from MarkFL and JDS
    Follow Math Help Forum on Facebook and Google+

  4. #4
    JDS
    JDS is offline
    Member
    Joined
    Oct 2012
    From
    South Carolina
    Posts
    83

    Re: Evaluate the derivative using properties of logarithms where needed.

    Thanks for the quick reply! Here is what I came up with and hopefully I have not confused myself!

    = (1/x5 sin2x) * x5

    = 1/sin2x
    Follow Math Help Forum on Facebook and Google+

  5. #5
    JDS
    JDS is offline
    Member
    Joined
    Oct 2012
    From
    South Carolina
    Posts
    83

    Re: Evaluate the derivative using properties of logarithms where needed.

    Quote Originally Posted by Prove It View Post
    Following Mark's example of simplifying the logarithm, you should simplify even further before trying to take the derivative. Write it as \displaystyle \begin{align*} y = 5\ln{(x)} + \ln{\left[ \sin{(x)} \right]} + \ln{\left[ \cos{(x)} \right]}  \end{align*} and then apply the much simpler chain rules to each term.
    By following that, I get....

    Y = 5 ln (x) + ln[sin(x)] + ln[cos(x)]

    5 ln (x) + ln[cos(x)] + ln[-sin(x)]

    grrr, and I am defintely sure I am confused now! LOL!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    1,988
    Thanks
    734

    Re: Evaluate the derivative using properties of logarithms where needed.

    Let's take a look at both approaches:

    My approach:

    \frac{d}{dx}(\ln(x^5\sin(2x)))=\frac{x^5(2\cos(2x)  )+5x^4\sin(2x))}{x^5\sin(2x)}=2\cot(2x)+5x^{-1}

    Prove It's approach:

    \frac{d}{dx}(5\ln(x)+\ln(\sin(x))+\ln(\cos(x)))=5x  ^{-1}+\frac{\cos(x)}{\sin(x)}-\frac{\sin(x)}{\cos(x)}=

    2\cot(2x)+5x^{-1}

    My approach has a more difficult application of the chain rule, but no need to apply double-angle identities (at least if you wish to combine the two trig. terms).
    Thanks from JDS
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: November 17th 2012, 10:51 AM
  2. properties of logarithms
    Posted in the Algebra Forum
    Replies: 4
    Last Post: February 24th 2010, 04:32 AM
  3. Logarithms Properties/Identities/Etc.
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: September 22nd 2009, 08:07 AM
  4. Properties of Logarithms
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: April 22nd 2009, 01:04 AM
  5. Properties of logarithms
    Posted in the Algebra Forum
    Replies: 3
    Last Post: March 11th 2007, 09:08 PM

Search Tags


/mathhelpforum @mathhelpforum